Regularized determinants of the Rumin complex in irreducible unitary representations of the (2,3,5) nilpotent Lie group

We study the Rumin differentials of the 5-dimensional graded nilpotent Lie group that appears as osculating group of generic rank two distributions in dimension five. In irreducible unitary representations of this group, the Rumin differentials provide intriguing generalizations of the classical quantum harmonic oscillator. For the Schrodinger representations, we compute the spectrum and the zeta regularized determinant of each Rumin differential. In the generic representations, we evaluate their alternating product, i.e., the analytic torsion of the Rumin complex.